Title:Coupling proofs are probabilistic product programs

Abstract: Couplings are a powerful mathematical tool for reasoning about pairs of
probabilistic processes. Recent developments in formal verification identify a
close connection between couplings and pRHL, a relational program logic
motivated by applications to provable security, enabling formal construction of
couplings from the probability theory literature. However, existing work using
pRHL merely shows existence of a coupling and does not give a way to prove
quantitative properties about the coupling, which are need to reason about
mixing and convergence of probabilistic processes. Furthermore, pRHL is
inherently incomplete, and is not able to capture some advanced forms of
couplings such as shift couplings. We address both problems as follows.
First, we define an extension of pRHL, called xpRHL, which explicitly
constructs the coupling in a pRHL derivation in the form of a probabilistic
product program that simulates two correlated runs of the original program.
Existing verification tools for probabilistic programs can then be directly
applied to the probabilistic product to prove quantitative properties of the
coupling. Second, we equip pRHL with a new rule for while loops, where
reasoning can freely mix synchronized and unsynchronized loop iterations. Our
proof rule can capture examples of shift couplings, and the logic is relatively
complete for deterministic programs.
We show soundness of xpRHL and use it to analyze two classes of examples.
First, we verify rapid mixing using different tools from coupling: standard
coupling, shift coupling, and path coupling, a compositional principle for
combining local couplings into a global coupling. Second, we verify
(approximate) equivalence between a source and an optimized program for several
instances of loop optimizations from the literature.